Semilinear Elliptic Systems Involving Multiple Critical Exponents And Singularities In R~N

In this paper, two kinds of semilinear elliptic systems inR^Nare investigated,which involve multiple critical exponents and singularities. First of all in theintroduction section, we introduce the two systems themselves and the relevant researchbackground, which makes us to know more about the problem. Meanwhile, we alsoelaborate some basic knowledge in this section, which makes preparations for the nextresearch.In the second chapter, we respectively establish the local Palais-Smale conditionsof the corresponding functionals by the concentration compactness principle. Since thetwo problems have multiple critical nonlinearities (In particular, the strongly coupledterms u^α-1 v^βand u^α v^β-1, and the corresponding functionals lose global compactness,we must establish the local Palais-Smale condition. On the other hand, the two systemsinvolve multiple singularities, and we need to study the problems inR^N, thus we usethe concentration compactness principle to prove the convergence of the criticalsequences.In the third chapter, we study the relationships of the best constants related to thetwo elliptic systems, which is an important premise for us to study the existence of theground states. According to the relationships between the best constants, we can get theextremal functions of the relevant best constants. Since the solutions which make theRayleigh quotient get minimum are the ground states of the system, we can completethe research of the ground states through the best constants.In the fourth chapter, we prove the existences of the ground states to the twosystems. By the local Palais-Smale, we can get the critical points of the correspondingenergy functional inR^N, and then we can prove the existence of the nonnegativeground states by the best constants. In order to get the positive ground states, we takethe Maximum principle. However, the systems we study are semilinear and we can only exclude the case that the nonnegative ground states is the point （0,0） by the Maximumprinciple, so we must get other methods to exclude the cases of nontrivial solutions.Finally in the fifth chapter, we study the nonexistences of the ground states to thetwo systems. In the process of research, we mainly argue by contradiction and employthe relationships between the best constants.